Barkhausen stability criterion

For the noise in the output of a ferromagnet upon a change in the magnetizing force, see Barkhausen effect.

Block diagram of a feedback oscillator circuit to which the Barkhausen criterion applies. It consists of an amplifying element A whose output vo is fed back into its input vf through a feedback network β(jω).

To find the loop gain, the feedback loop is considered broken at some point and the output vo for a given input vi is calculated:
:

It states that if A is the gain of the amplifying element in the circuit and β(jω) is the transfer function of the feedback path, so βA is the loop gain around the feedback loop of the circuit, the circuit will sustain steady-state oscillations only at frequencies for which:

Barkhausen's criterion is a necessary condition for oscillation but not a sufficient condition: some circuits satisfy the criterion but do not oscillate.[5] Similarly, the Nyquist stability criterion also indicates instability but is silent about oscillation. Apparently there is not a compact formulation of an oscillation criterion that is both necessary and sufficient.[6]

Barkhausen's original "formula for self-excitation", intended for determining the oscillation frequencies of the feedback loop, involved an equality sign: |βA| = 1. At the time conditionally-stable nonlinear systems were poorly understood; it was widely believed that this gave the boundary between stability (|βA| < 1) and instability (|βA| ≥ 1), and this erroneous version found its way into the literature.[7] However, stable oscillations only occur at frequencies for which equality holds.